Use of Kalman filters in time and
frequency analysis
John Davis
1st May 2011
Overview of Talk
 Simple example of Kalman filter
 Discuss components and operation of filter
 Models of clock and time transfer noise and
deterministic properties
 Examples of use of Kalman filter in time and
frequency
Predictors, Filters and Smoothing Algorithms
 Predictor: predicts parameter values ahead of current
measurements
 Filter: estimates parameter values using current and
previous measurements
 Smoothing Algorithm: estimates parameter values
using future, current and previous measurements
What is a Kalman Filter ?
 A Kalman filter is an optimal recursive data
processing algorithm
 The Kalman filter incorporates all information that can
be provided to it. It processes all available
measurements, regardless of their precision, to
estimate the current value of the variables of interest
 Computationally efficient due to its recursive structure
 Assumes that variables being estimated are time
dependent
 Linear Algebra application
Simple Example
1.5E-08
1.0E-08
Offset
5.0E-09
0.0E+00
0
2
4
6
8
10
12
-5.0E-09
-1.0E-08
Date
Truth
Measurement
14
16
18
20
Simple Example (Including Estimates)
1.5E-08
1.0E-08
5.0E-09
0.0E+00
0
2
4
6
8
10
12
14
-5.0E-09
-1.0E-08
Truth
Measurement
Filter Estimate
16
18
20
Estimation Errors and Uncertainty
6.0E-09
4.0E-09
Error
2.0E-09
0.0E+00
0
2
4
6
8
10
12
14
-2.0E-09
-4.0E-09
-6.0E-09
Date
Error
Uncertainty (lower)
Uncertainty (upper)
16
18
20
Operation of Kalman Filter
 Provides an estimate of the current parameters using
current measurements and previous parameter
estimates
 Should provide a close to optimal estimate if the
models used in the filter match the physical situation
 World is full of badly designed Kalman filters
What are the applications in time and frequency?
 Analysis of time transfer / clock measurements where
measurement noise is significant
 Clock and time scale predictors
 Clock ensemble algorithms
 Estimating noise processes
 Clock and UTC steering algorithms
 GNSS applications
Starting point for iteration 11 of filter
1.4E-08
1.2E-08
1.0E-08
Offset
8.0E-09
6.0E-09
4.0E-09
2.0E-09
0.0E+00
-2.0E-09
0
2
4
6
8
10
12
Date
Measurement
Filter Estimate
14
16
18
20
Five steps in the operation of a Kalman filter
 (1) State Vector propagation
 (2) Parameter Covariance Matrix propagation
 (3) Compute Kalman Gain
 (4) State Vector update
 (5) Parameter Covariance Matrix update
Components of the Kalman Filter (1)
State vector x
 Contains required parameters. These parameters
usually contain deterministic and stochastic
components.
 Normally include: time offset, normalised frequency
offset, and linear frequency drift between two clocks
or timescales.
 State vector components may also include: Markov
processes where we have memory within noise
processes and also components of periodic
instabilities
Components of the Kalman Filter (2)
State Propagation Matrix F(t)
 Matrix used to extrapolate the deterministic
characteristics of the state vector from time t to t+t
Steps in the operation of the Kalman Filter
Step 1: State vector propagation from iteration n-1 to n

xˆ (tn )  F(t ) xˆ (tn1 )


xˆ (tn ) Estimate of the state vector at iteration n not
including measurement n.
F(t )
spacing t
State propagation matrix calculated for time

xˆ (tn1 ) Estimate of the state vector at iteration n-1
including measurement n-1.
Step 1: State Vector Extrapolation
1.4E-08
1.2E-08
1.0E-08
Offset
8.0E-09
6.0E-09
4.0E-09
2.0E-09
0.0E+00
0
2
4
6
8
10
12
14
-2.0E-09
Date
Measurement
Filter Estimate
State Vector Extrapolation
16
18
20
Components of the Kalman Filter (3)
Parameter Covariance Matrix P
 Contains estimates of the uncertainty and correlation
between uncertainties of state vector components.
Based on information supplied to the Kalman filter.
Not obtained from measurements.
Process Covariance Matrix Q(t)
 Matrix describing instabilities of components of the
state vector, e.g. clock noise, time transfer noise
moving from time t to t+t
Noise parameters s
 Parameters used to determine the elements of the
process covariance matrix, describe individual noise
processes.
Steps in the operation of the Kalman Filter
Step 2: Parameter covariance matrix propagation from
iteration n-1 to n

P (tn )  F(t ) P (tn1 )F (t )  Q(tn )

T
Parameter covariance matrix, at iteration n1 including measurement n-1
T
F(t ) F (t ) State propagation matrix and
transpose calculated for time spacing t

P (tn ) Parameter covariance matrix, at iteration n
not including measurement n
Q(tn ) Process covariance matrix
Uncertainty Extrapolation
6.0E-09
Error / Uncertainty
4.0E-09
2.0E-09
0.0E+00
0
2
4
6
8
10
12
14
16
18
-2.0E-09
-4.0E-09
-6.0E-09
Error
Uncertainty (upper)
Date
Uncertainty Extrapolation (upper)
Uncertainty (lower)
Uncertainty Extrapolation (lower)
20
Components of the Kalman Filter (4)
Design Matrix H
 Matrix that relates the measurement vector and the
state vector using y =H x
Measurement Covariance Matrix R
 Describes measurement noise, white but individual
measurements may be correlated. May be removed.
Kalman Gain K
 Determines the weighting of current measurements and
estimates from previous iteration. Computed by filter.
Steps in the operation of the Kalman Filter
Step 3 :Kalman gain computation



K (tn )  P (tn ) H H P (tn ) H  R(tn )
T
T


1
P (tn ) Parameter covariance matrix, at iteration n not
including measurement n
T
H H Design Matrix and Transpose
K (tn ) Kalman Gain
R(tn ) Measurement covariance matrix
Components of the Kalman Filter (5)
Measurement Vector y
 Vector containing measurements input during a
single iteration of the filter
Identity Matrix I
Steps in the operation of the Kalman Filter
Step 4 :State vector update




xˆ (tn )  xˆ (tn )  K (tn ) y(tn )  H xˆ (tn )


xˆ (tn ) xˆ (tn ) Estimate of the state vector at iteration n
including and not including measurement n respectively.
K (tn ) Kalman Gain
y (tn ) Measurement Vector
H Design Matrix

Step 4 :State Vector Update
1.4E-08
1.2E-08
1.0E-08
Offset
8.0E-09
6.0E-09
4.0E-09
2.0E-09
0.0E+00
0
2
4
6
8
10
12
14
16
-2.0E-09
Measurement
Filter Estimate
Date
State Vector Extrapolation
State Vector Update
18
20
Steps in the operation of the Kalman Filter
Step 5 :Parameter Covariance Matrix update

n



n
P(t )  I  K (tn ) H P(t )
Kalman Gain
H Design Matrix
P(tn ) P(tn ) Parameter Covariance Matrix, at iteration
n, including and not including measurement n
respectively
I Identity Matrix
K (tn )
Steps in the operation of the Kalman Filter
If Kalman gain is deliberately set sub-optimal, then use
P(t )  I  K (tn ) H P(t )I  K (tn ) H 

n

n
Kalman Gain
H Design Matrix
P(tn ) P(tn ) Parameter Covariance Matrix, at
iteration n, including and not including measurement
n respectively
I Identity Matrix
K (tn )
T
Step 5: Uncertainty Update
6.0E-09
Error / Uncertainty
4.0E-09
2.0E-09
0.0E+00
0
2
4
6
8
10
12
14
16
18
-2.0E-09
-4.0E-09
-6.0E-09
Date
Error
Uncertainty Extrapolation (lower)
Uncertainty update (upper)
Uncertainty (lower)
Uncertainty Extrapolation (upper)
Uncertainty (upper)
Uncertainty Update (lower)
20
Constructing a Kalman Filter:
Lots to think about!










Choice of State Vector components
Choice of Noise Parameters
Choice of measurements
Description of measurement noise
Choice of Design Matrix
Choice of State Propagation Matrix
Initial set up conditions
Dealing with missing data
Testing out the filter
Computational methods
Initialising the Kalman Filter
 Set the initial parameters to physically realistic values
 If you have no information set diagonal values of
Parameter Covariance Matrix high, filter will then give a
high weight to the first measurement.
 Set Parameters with known variances e.g. Markov
components of Parameter Covariance Matrices to their
steady state values.
 Set Markov State Vector components and periodic
elements to zero.
 Elements of H, F(t), Q(t), R are pre-determined
Simulating Noise and Deterministic Processes
 Use the same construction of state vector
components x, state propagation matrix F(t),
parameter covariance matrix P, process covariance
matrix Q(t), design matrix H and measurement
covariance matrix R.
 Simulate all required data sets
 Simulate “true” values of state vector for error
analysis
 May determine ADEV, HDEV, MDEV and TDEV
statistics from parameter covariance matrix
Simulating Noise and Deterministic Processes
x(tn )  F (t ) x(tn1 )  L Q (tn )e(tn )
Where
Q  LQ (tn ) LQ (tn )
T
Q = Process Covariance Matrix
x(tn ) x(tn1 ) = State Vector at iteration n and n-1
respectively
F(t ) = State Propagation Matrix calculated for time
spacing t
e(tn )  Vector of normal distribution noise of unity
magnitude
Simulating Noise and Deterministic Processes
P(tn )  F(t ) P(tn1 )F (t )  Q(tn )
T
P(tn ) P(tn1 ) Process Covariance Matrix at iterations n
and n-1 respectively
F(t ) F (t ) State Propagation Matrix and
T
transpose computed at time spacing t.
Q (tn ) Process Covariance Matrix calculated at time
spacing t
Simulating Noise and Deterministic Processes
y (tn )  H x(tn )  LR (tn )e(tn )
where
y(tn )
H
R
e(tn )
R  LR (tn ) L (tn )
T
R
Measurement Vector
Design Matrix
Measurement Covariance Matrix
Vector of normal distribution noise of unity
magnitude
ADEV, HDEV, and MDEV determined
from Parameter Covariance Matrix and simulation
10
-12
ADEV simulation
HDEV simulation
MDEV simulation
ADEV theory
HDEV theory
MDEV theory
z
-13
Sigma
10
10
10
-14
-15
10
2
10
3
10
4
10
tau
5
10
6
10
7
Simulating Noise and Deterministic Processes
 Test Kalman filter with a data set with same
stochastic and deterministic properties as is expected
by the Kalman filter.
 Very near optimal performance of Kalman filter under
these conditions.
Deterministic Properties
 Linear frequency drift
 Use three state vector components, time offset,
normalised frequency offset and linear frequency drift
 1 t t 2 2 


t
0 1

F

0
0
1






Markov Noise Process




x e
White Noise
n
Random Walk Noise
n
n1
Markov Noise
0  k 1
n
n1
Flicker noise, combination of Markov processes with
continuous range of relaxation parameters k
 Markov noise processes and flicker noise have
memory
 Kalman filters tend to work very well with Markov
noise processes
 May only construct an approximation to flicker noise
in a Kalman filter
x  x e
x  kx  e
Markov Noise Process
6
x 10
-6
4
Time Offset
2
0
-2
-4
-6
6
6.5
7
MJD
7.5
8
Describing Measurement Noise
 Kalman filter assumes measurement noise
introduced via matrix R is white
 Time transfer noise e.g TWSTFT and GNSS noise is
often far from white
 Add extra components to state vector to describe non
white noise, particularly useful if noise has “memory”
e.g. flicker phase noise
 NPL has not had any problems in not using a
measurement noise matrix R
White Phase Modulation
x 10
WPM Noise
-10
Time Offset
0.5
0
-0.5
0
5
10
MJD
15
20
White Phase Modulation
8
x 10
-11
WPM Noise (sample)
6
Time Offset
4
2
0
-2
-4
10
10.2
10.4
10.6
MJD
10.8
11
White Phase Modulation
10
10
Sigma
10
10
10
10
ADEV, HDEV MDEV of WPM
-12
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
-13
-14
-15
-16
-17
10
2
10
3
4
10
tau
10
5
10
6
Flicker Phase Modulation
2
x 10
-10
Simulated FPM Noise
1.5
1
Time Offset
0.5
0
-0.5
-1
-1.5
-2
-2.5
0
5
10
MJD
15
20
Flicker Phase Modulation
10
Sigma
10
10
10
10
ADEV, HDEV MDEV of FFM
-12
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
-13
-14
-15
-16
10
2
10
3
4
10
tau
10
5
10
6
Measurement noise used in Kalman filter
2
x 10
Simulated Time Transfer Noise
-10
Time Offset
1
0
-1
-2
0
5
10
MJD
15
20
Measurement noise ADEV, MDEV and HDEV
ADEV,
HDEV MDEV of measurements noise
-12
10
Sigma
10
10
10
10
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
-13
-14
-15
-16
10
2
10
3
4
10
tau
10
5
10
6
Clock Noise
 White Frequency Modulation and Random Walk
Frequency Modulation easy to include as single noise
parameters.
 Flicker Frequency Modulation may be modelled as
linear combination of Integrated Markov Noise
Parameters
White Frequency Modulation
5
x 10
WFM Noise
-8
Time Offset
0
-5
-10
-15
0
5
10
MJD
15
20
White Frequency Modulation
10
ADEV, HDEV MDEV of WFM
-11
Sigma
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
10
10
-12
-13
10
2
10
3
4
10
tau
10
5
10
6
Random Walk Frequency Modulation
0
x 10
RWFM Noise
-7
-1
-2
Time Offset
-3
-4
-5
-6
-7
-8
-9
0
5
10
MJD
15
20
Random Walk Frequency Modulation
10
-13
Sigma
10
ADEV, HDEV MDEV of RWFM
-12
10
10
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
-14
-15
10
2
10
3
4
10
tau
10
5
10
6
FFM Noise
2
x 10
FFM Noise
-9
1
0
Time Offset
-1
-2
-3
-4
-5
-6
-7
0
5
10
MJD
15
20
Flicker Frequency Modulation
ADEV, HDEV MDEV of FFM noise
-14
Sigma
10
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
10
-15
10
2
10
3
4
10
tau
10
5
10
6
Periodic Instabilities





Often occur in GNSS Applications
Orbit mis-modelling
Diurnal effects on Time Transfer hardware
Periodicity decays over time
Model as combination of Markov noise process and
periodic parameter
 Two extra state vector component to represent
periodic variations
GPS Time – Individual Clock (Real Data)
Input data space clock prn 9
-5
1.622
x 10
Input Data
1.6215
Time Offset
1.621
1.6205
1.62
1.6195
1.619
0
1
2
3
4
Day
D
5
6
7
GPS Time – Individual Clock (Simulated Data)
-8
Simulated
data with similar characteristics
x 10
1
GPS CV
Time Offset
0.5
0
-0.5
-1
-1.5
6
7
8
9
10
MJD
11
12
13
ADEV, HDEV, MDEV Real Data
10
ADEV / HDEV / MDEV Prn 9
-11
ADEV / HDEV / MDEV
ADEV
HDEV
MDEV
10
10
10
-12
-13
-14
10
2
10
3
4
10
Tau
10
5
10
6
ADEV, HDEV, MDEV Simulated Data
ADEV, HDEV MDEV Theory and Measurement
10
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
-12
ADEV
10
-11
10
10
-13
-14
10
2
10
3
4
10
tau
10
5
10
6
Kalman Filter Residuals

( y  Hxˆ )
 Residuals
 Residual Variance (Theory)


V  HP H
T

 In an optimally constructed Kalman Filter the
Residuals should be white!
 Residual Variance obtained from Parameter
Covariance matrix should match that obtained from
statistics applied to residuals
Kalman Filter Residuals (Simulated Data)
-9
Kalman
Filter Residuals (simulated data)
x 10
1
Time Offset
0.5
0
-0.5
-1
-1.5
5
5.5
6
6.5
MJD
7
7.5
8
Kalman Filter Residuals (Real Data)
-9
Kalman
Filter Residuals (real data)
1.5
x 10
Time Offset
1
0.5
0
-0.5
-1
-1.5
2
2.5
3
3.5
MJD
4
4.5
5
Computational Issues
 Too many state vector components result in a significant
computational overhead
 Observability of state vector components
 Direct computation vs factorisation
 Direct use of Kalman filter equations usually numerically stable and
easy to follow.
 Issue of RRFM scaling if a linear frequency drift state vector
component is used
Clock Predictor Application
 May provide a close to optimal prediction in an
environment that contains complex noise and
deterministic characteristics
 If noise processes are simple e.g. WFM and no linear
frequency drift then very simple predictors will work
very well.
Clock Predictor Equations (Prediction)
xˆ (t )  F(t ) xˆ (t

n

n 1
)
• Repeated application of above equation.
•Apply above equations with F (t ) calculated
at the required prediction length t
Clock Predictor Equations (PED)
P(t )  F(t ) P(t )F (t )  Q(tn )

n

n1
T
 Repeated application of above equation.
 Apply above equations with
required prediction length t
F (t ) calculated at the
Example: Predicting clock offset in presence
of periodic instabilities

ADEV = PED/Tau
 Above equation exact in case of WFM and RWFM
noise processes, and a reasonable approximation in
other situations
ADEV, HDEV, MDEV Simulated Data
ADEV, HDEV MDEV Theory and Measurement
10
ADEV Measurements
HDEV Measurements
MDEV Measurements
ADEV Theory
HDEV Theory
MDEV Theory
-12
ADEV
10
-11
10
10
-13
-14
10
2
10
3
4
10
tau
10
5
10
6
(PED /Tau) Prediction of Space Clocks
(simulated data)
M ean P E D / tau
-1 1 .4
P re d ictio n D e via tio n C lo ck a nd No ise
P re d ictio n E rro r
-1 1 .6
-1 1 .8
log P E D / tau
-1 2
-1 2 .2
-1 2 .4
-1 2 .6
-1 2 .8
-1 3
-1 3 .2
-1 3 .4
2
2 .5
3
3 .5
4
lo g Ta u
4 .5
5
5 .5
ADEV, HDEV, MDEV Real Data
10
ADEV / HDEV / MDEV Prn 9
-11
ADEV / HDEV / MDEV
ADEV
HDEV
MDEV
10
10
10
-12
-13
-14
10
2
10
3
4
10
Tau
10
5
10
6
(PED /Tau) Prediction of Space Clocks
(real data)
M e a n P E D / ta u
-1 1 .4
P re d ic tio n D e via tio n C lo c k a nd N o is e
P re d ic tio n E rro r
 Add plot
-1 1 .6
lo g P E D / ta u
-1 1 .8
-1 2
-1 2 .2
-1 2 .4
-1 2 .6
-1 2 .8
-1 3
-1 3 .2
-1 3 .4
2
2 .5
3
3 .5
4
lo g Ta u
4 .5
5
5 .5
Kalman Filter Ensemble Algorithm
 Most famous time and frequency application of the
Kalman filter
 Initially assume measurements are noiseless.
 Measurements (Cj – C1), difference between pairs of
individual clocks
 Estimates (T – Ci), where T is a “perfect” clock.
 Parameters being estimated cannot be constructed
from the measurements:
 Badly designed Kalman filter.
 Problem solved using covariance x reduction
Stability of Composite Timescales
-14.2
Clock 1 (FFM)
Clock 2 (WFM)
Clock 3 (WFM)
Composite
Simple Composite
Optimal Composite
-14.4
-14.6
Log10(sy)
-14.8
-15
-15.2
-15.4
-15.6
-15.8
2
2.5
3
3.5
Log10(t)
4
4.5
5
Advantage of Kalman Filter based Ensemble
Algorithm
 Potential of close to optimal performance over a range
of averaging times
 Clasical Ensemble Algorithms provide a very close to
optimal performance at a single averaging time
Combining TWSTFT and GPS Common-View
measurements
 Time transfer noise modelled using Markov Noise
processes
 Calibration biases modelled using long relaxation
time Markov
Real Time Transfer Data Example
6.5
x 10
-8
Measurements actually used
GPS CV
TWSTFT
6
Time-Offset (s)
5.5
5
4.5
4
3.5
3
2.5
5.4405
5.441
5.4415
MJD
5.442
5.4425
x 10
4
Kalman Filter Estimates
5.5
x 10
-8
True time-offset and estimates
Estimated Time-Offset
Time-Offset (s)
5
4.5
4
3.5
5.4405
5.441
5.4415
MJD
5.442
5.4425
x 10
4
Kalman Filter Residuals
8
x 10
-9
Kalman Filter Residuals
GPS CV
TWSTFT
6
4
Time-Offset (s)
2
0
-2
-4
-6
-8
-10
-12
5.4405
5.441
5.4415
MJD
5.442
5.4425
x 10
4
Summary
 Operation of the Kalman filter
 Stochastic and deterministic models for time and
frequency applications
 Real examples of use Kalman Filter
Title of Presentation
Name of Speaker
DateThe National Measurement System delivers world-class
measurement science & technology through these organisations
The National Measurement System is the UK’s national infrastructure of measurement
Laboratories, which deliver world-class measurement science and technology through four
National Measurement Institutes (NMIs): LGC, NPL the National Physical Laboratory, TUV NEL
The former National Engineering Laboratory, and the National Measurement Office (NMO).